arXiv:1207.3614 · Physical Review A

N -dimensional alternate coined quantum walks from a dispersion-relation perspective

Eugenio Roldán, Carlo D. Franco, Fernando Silva, Germán J. de Valcárcel

Published Feb 25, 2013• 16 citations

Abstract

We suggest an alternative definition of $N$-dimensional coined quantum walk by generalizing a recent proposal [Di Franco et al., Phys. Rev. Lett. 106, 080502 (2011)]. This $N$-dimensional alternate quantum walk, ${\mathrm{AQW}}_{(N)}$, in contrast with the standard definition of the $N$-dimensional quantum walk, ${\mathrm{QW}}_{(N)}$, requires only a coin qubit. We discuss the quantum diffusion properties of ${\mathrm{AQW}}_{(2)}$ and ${\mathrm{AQW}}_{(3)}$ by analyzing their dispersion relations that reveal, in particular, the existence of diabolical points. This allows us to highlight interesting similarities with other well-known physical phenomena. We also demonstrate that ${\mathrm{AQW}}_{(3)}$ generates considerable genuine multipartite entanglement. Finally, we discuss the implementability of ${\mathrm{AQW}}_{(N)}$.

DOI: 10.1103/physreva.87.022336

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arXiv:1207.3614 · Physical Review A

N -dimensional alternate coined quantum walks from a dispersion-relation perspective

Eugenio Roldán, Carlo D. Franco, Fernando Silva, Germán J. de Valcárcel

Published Feb 25, 2013• 16 citations

arXiv HTML PDF

Abstract

We suggest an alternative definition of

N

-dimensional coined quantum walk by generalizing a recent proposal [Di Franco et al., Phys. Rev. Lett. 106, 080502 (2011)]. This

N

-dimensional alternate quantum walk,

{\mathrm{AQW}}_{(N)}

, in contrast with the standard definition of the

N

-dimensional quantum walk,

{\mathrm{QW}}_{(N)}

, requires only a coin qubit. We discuss the quantum diffusion properties of

{\mathrm{AQW}}_{(2)}

and

{\mathrm{AQW}}_{(3)}

by analyzing their dispersion relations that reveal, in particular, the existence of diabolical points. This allows us to highlight interesting similarities with other well-known physical phenomena. We also demonstrate that

{\mathrm{AQW}}_{(3)}

generates considerable genuine multipartite entanglement. Finally, we discuss the implementability of

{\mathrm{AQW}}_{(N)}

DOI: 10.1103/physreva.87.022336

Related papers

Alternate two-dimensional quantum walk with a single-qubit coin

Carlo D. Franco, Michael Mc Gettrick, Takuya Machida +1 more

We have recently proposed a two-dimensional quantum walk where the requirement of a higher dimensionality of the coin space is substituted with the alternance of the directions in which the walker can move [C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011)]. For a particular initial state of the coin, this walk is able to perfectly reproduce the spatial probability distribution of the nonlocalized case of the Grover walk. Here, we present a more detailed proof of this equivalence. We also extend the analysis to other initial states in order to provide a more complete picture of our walk. We show that this scheme outperforms the Grover walk in the generation of $x$ - $y$ spatial entanglement for any initial condition, with the maximum entanglement obtained in the case of the particular aforementioned state. Finally, the equivalence is generalized to wider classes of quantum walks and a limit theorem for the alternate walk in this context is presented.We have recently proposed a two-dimensional quantum walk where the requirement of a higher dimensionality of the coin space is substituted with the alternance of the directions in which the walker can move [C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011)]. For a particular initial state of the coin, this walk is able to perfectly reproduce the spatial probability distribution of the nonlocalized case of the Grover walk. Here, we present a more detailed proof of this equivalence. We also extend the analysis to other initial states in order to provide a more complete picture of our walk. We show that this scheme outperforms the Grover walk in the generation of $x$-$y$ spatial entanglement for any initial condition, with the maximum entanglement obtained in the case of the particular aforementioned state. Finally, the equivalence is generalized to wider classes of quantum walks and a limit theorem for the alternate walk in this context is presented.

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The discrete quantum walk in N dimensions is analyzed from the perspective of its dispersion relations. This allows understanding known properties, as well as designing new ones when spatially extended initial conditions are considered. This is done by deriving wave equations in the continuum, which are generically of the Schrödinger type, and allows devising interesting behavior, such as ballistic propagation without deformation, or the generation of almost flat probability distributions, which is corroborated numerically. There are however special points where the energy surfaces display intersections and, near them, the dynamics is entirely different. Applications to the two- and three-dimensional Grover walks are presented.The discrete quantum walk in N dimensions is analyzed from the perspective of its dispersion relations. This allows understanding known properties, as well as designing new ones when spatially extended initial conditions are considered. This is done by deriving wave equations in the continuum, which are generically of the Schrödinger type, and allows devising interesting behavior, such as ballistic propagation without deformation, or the generation of almost flat probability distributions, which is corroborated numerically. There are however special points where the energy surfaces display intersections and, near them, the dynamics is entirely different. Applications to the two- and three-dimensional Grover walks are presented.

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One of the proposals for the exploitation of two-dimensional quantum walks has been the efficient generation of entanglement. Unfortunately, the technological effort required for the experimental realization of standard two-dimensional quantum walks is significantly demanding. In this respect, an alternative scheme with less challenging conditions has been recently studied, particularly in terms of spatial-entanglement generation [C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011)]. Here, we extend the investigation to a scenario where a measurement is performed on the coin degree of freedom after the evolution, allowing a further comparison with the standard two-dimensional Grover walk.One of the proposals for the exploitation of two-dimensional quantum walks has been the efficient generation of entanglement. Unfortunately, the technological effort required for the experimental realization of standard two-dimensional quantum walks is significantly demanding. In this respect, an alternative scheme with less challenging conditions has been recently studied, particularly in terms of spatial-entanglement generation [C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011)]. Here, we extend the investigation to a scenario where a measurement is performed on the coin degree of freedom after the evolution, allowing a further comparison with the standard two-dimensional Grover walk.

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A coinless, discrete-time quantum walk possesses a Hilbert space whose dimension is smaller compared to the widely studied coined walk. Coined walks require the direct product of the site basis with the coin space; coinless walks operate purely in the site basis, which is clearly minimal. These coinless quantum walks have received considerable attention recently because they have evolution operators that can be obtained by a graphical method based on lattice tessellations and they have been shown to be as efficient as the best known coined walks when used as a quantum search algorithm. We argue that both formulations in their most general form are equivalent. In particular, we demonstrate how to transform the one-dimensional version of the coinless quantum walk into an equivalent extended coined version for a specific family of evolution operators. We present some of its basic, asymptotic features for the one-dimensional lattice with some examples of tessellations, and analyze the mixing time and limiting probability distributions on cycles.A coinless, discrete-time quantum walk possesses a Hilbert space whose dimension is smaller compared to the widely studied coined walk. Coined walks require the direct product of the site basis with the coin space; coinless walks operate purely in the site basis, which is clearly minimal. These coinless quantum walks have received considerable attention recently because they have evolution operators that can be obtained by a graphical method based on lattice tessellations and they have been shown to be as efficient as the best known coined walks when used as a quantum search algorithm. We argue that both formulations in their most general form are equivalent. In particular, we demonstrate how to transform the one-dimensional version of the coinless quantum walk into an equivalent extended coined version for a specific family of evolution operators. We present some of its basic, asymptotic features for the one-dimensional lattice with some examples of tessellations, and analyze the mixing time and limiting probability distributions on cycles.

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Classification of coined quantum walks on the line and comparison to correlated classical random walks

Lukas Hantzko, Lennart Binkowski

Abstract We present a comprehensive classification of one-dimensional coined quantum walks on the infinite line, focusing on the spatial probability distributions they induce. Building on prior results, we identify all initial coin states that lead to symmetric quantum walks for arbitrary coins and provide a bijective parametrisation of all symmetric quantum walks modulo distributional equivalence. Extending beyond the symmetric case, we also give a surjective parametrisation of all coined quantum walks under the same equivalence relation, as well as a bijective parametrisation modulo equivalence of their limiting distributions. Furthermore, we derive corrected closed-form expressions for the walk amplitudes, resolving inaccuracies in previous literature, and generalise the approach to the correlated classical random walk. This unified framework enables a direct comparison between quantum and classical dynamics. Additionally, we discuss the asymptotic scaling of variances for both models, identifying quadratic spreading as a hallmark of non-trivial quantum walks and contrasting it with the linear behaviour of classical walks, except at the extremal points of maximal correlation. Finally, we compare the limiting distributions arising from quantum walks with those of the classical case. Graphical abstract - compact classification results for symmetric and general quantum walks, as well as classically correlated random walks - quantum walks spread ballistically while classically correlated random walks approach Gaussian limiting distribution - quantum coin parameter directly influence limiting distribution’s skewnessAbstract We present a comprehensive classification of one-dimensional coined quantum walks on the infinite line, focusing on the spatial probability distributions they induce. Building on prior results, we identify all initial coin states that lead to symmetric quantum walks for arbitrary coins and provide a bijective parametrisation of all symmetric quantum walks modulo distributional equivalence. Extending beyond the symmetric case, we also give a surjective parametrisation of all coined quantum walks under the same equivalence relation, as well as a bijective parametrisation modulo equivalence of their limiting distributions. Furthermore, we derive corrected closed-form expressions for the walk amplitudes, resolving inaccuracies in previous literature, and generalise the approach to the correlated classical random walk. This unified framework enables a direct comparison between quantum and classical dynamics. Additionally, we discuss the asymptotic scaling of variances for both models, identifying quadratic spreading as a hallmark of non-trivial quantum walks and contrasting it with the linear behaviour of classical walks, except at the extremal points of maximal correlation. Finally, we compare the limiting distributions arising from quantum walks with those of the classical case. Graphical abstract - compact classification results for symmetric and general quantum walks, as well as classically correlated random walks - quantum walks spread ballistically while classically correlated random walks approach Gaussian limiting distribution - quantum coin parameter directly influence limiting distribution’s skewness

arXiv:2507.23524The European Physical Journal DJul 1, 20260 citations

We suggest an alternative definition of

N

-dimensional coined quantum walk by generalizing a recent proposal [Di Franco et al., Phys. Rev. Lett. 106, 080502 (2011)]. This

N

-dimensional alternate quantum walk,

{\mathrm{AQW}}_{(N)}

, in contrast with the standard definition of the

N

-dimensional quantum walk,

{\mathrm{QW}}_{(N)}

, requires only a coin qubit. We discuss the quantum diffusion properties of

{\mathrm{AQW}}_{(2)}

and

{\mathrm{AQW}}_{(3)}

{\mathrm{AQW}}_{(3)}

generates considerable genuine multipartite entanglement. Finally, we discuss the implementability of

{\mathrm{AQW}}_{(N)}

x

y

spatial entanglement for any initial condition, with the maximum entanglement obtained in the case of the particular aforementioned state. Finally, the equivalence is generalized to wider classes of quantum walks and a limit theorem for the alternate walk in this context is presented.